Dispersion of the intrinsic neuronal periods affects the relationship of the entrainment range to the coupling strength in the suprachiasmatic nucleus

Living beings on the Earth are subjected to and entrained (synchronized) to the natural 24-h light-dark cycle. Interestingly, they can also be entrained to an external artificial cycle of non-24-h periods. The range of these periods is called the entrainment range and it differs among species. In mammals, the entrainment range is regulated by a main clock located in the suprachiasmatic nucleus (SCN) which is composed of 10 000 neurons in the brain. Previous works have found that the entrainment range depends on the cellular coupling strength in the SCN. In particular, the entrainment range decreases with the increase of the cellular coupling strength, provided that all the neuronal oscillators are identical. However, the SCN neurons differ in the intrinsic periods that follow a normal distribution in a range from 22 to 28 h. In the present study, taking the dispersion of the intrinsic neuronal periods into account, we examined the relationship between the entrainment range and the coupling strength. Results from numerical simulations and theoretical analyses both show that the relationship is altered to be paraboliclike if the intrinsic neuronal periods are nonidentical, and the maximal entrainment range is obtained with a suitable coupling strength. Our results shed light on the role of the cellular coupling in the entrainment ability of the SCN network.

Figure 1

The relationship of the free running period ($\text{FRP}$) to the coupling strength $g$ [(a)–(c)] and the relationship of $\text{FRP}$ to the intrinsic neuronal amplitude $a$ [(d)–(f)] in the absence of a zeitgeber ($K_f=0$). The parameter $\delta$ represents the standard deviation of intrinsic neuronal periods, and the error bar measures the standard deviation of “coupled” neuronal periods.

Figure 2

Temporal evolutions of four randomly selected oscillators exposed to an external $T=23$ h cycle, when the coupling strength is $g=0$ (a) and $g=0.05$ (b). The dispersion is $\delta =0.05$. The gray regions correspond to darkness and the white regions to brightness.

Figure 3

The relationship of the LLE to the coupling strength $g$. The intrinsic neuronal amplitude is $a=0.5$ (a), $a=1.0$ (b), and $a=2.0$ (c), respectively. LLE represents the lower limit of entrainment, and $\delta$ represents the dispersion of the intrinsic neuronal periods. Because the larger LLE corresponds to the narrower entrainment range, the direction of the $y$ axis is reversed; the value of the LLE is absent when the coupling strength is weak ($g\le 0.02$) for $\delta =0.05$, because the SCN loses the entrainment to any external cycle (c).

Figure 4

The relationship of the LLE to the intrinsic neuronal amplitude $a$. The coupling strength is $g=0.02$ (a), $g=0.05$ (b), and $g=0.10$ (c), respectively. LLE represents the lower limit of entrainment, and $\delta$ represents the dispersion of the intrinsic neuronal periods. Please note that the direction of the $y$ axis is reversed, because the larger LLE corresponds to the narrower entrainment range.

Figure 5

The relationship of the LLE to the coupling strength $g$ in three cases: When the DM runs faster than the VL (a), when the VL dominates the DM (b), and when the SCN is a small-world network. Except that the average intrinsic frequency of the DM neurons is larger than the VL, i.e., ${\varpi }_{\text{DM}}=0.2635$ and ${\varpi }_{\text{VL}}=0.2566$, all the parameters are the same for these two subgroups for (a)–(c). The details of simulations for each case can be found in Sec. 2.

Figure 6

The relationship of the LLE to the coupling strength $g$ with different values of deviation $\delta$. The relationship is positive with $\delta =0$ and is paraboliclike with $\delta =0.03$ or 0.05. The parameters are $\gamma =0.2$, $N=2$, and $K_f=0.05$.

Figure 7

The effects of the coupling strength $g$ on the network properties, the entrained amplitudes $r_{\text{ent}}$ under the LLE (a), the phase differences under the LLE (b), the amplitude of the coupled system $r_{\text{cou}}$ under the constant darkness (c), and the ratio $\frac{r_{\text{ent}}}{r_{\text{cou}}}$ (d). The symbols $O_1$, $O_2$, and $S$ represent the DM, the VL, and the SCN network, respectively. ${\varphi }_1$, ${\varphi }_2$, and $\alpha$ represent the phase differences between the DM and the external cycle, between the VL and the external cycle, and between the VL and the DM, respectively. The amplitude of the network is defined as the mean of the amplitudes of the two oscillators. The parameters are $\gamma =0.2$, $N=2$, and $K_f=0.05$.

Figure 8

The effects of the coupling strength $g$ on the entrainment range [(a), (b)] and the effects of the neuronal amplitude $a$ on the entrainment range $\text{ER}$ [(c), (d)] when the neurons differ in the intrinsic periods. The parameters $S$, $r$, and $\text{ER}$ represent the degree of synchronization, coupled neuronal amplitude, and entrainment range, respectively.